> ## Content Index
> Fetch the complete content index at: https://starikov.co/llms.txt
> Use this file to discover other available public pages before exploring further.

# Painting Gabriel’s Horn
- URL: https://starikov.co/gabriels-horn/
- Published: 2016-10-10T18:11:00.000Z
- Updated: 2026-06-09T23:01:28.000Z
- Description: A horn you can fill with paint, but you can't paint the surface.
- Author: Illya Starikov
- Tags: Mathematics, #u-color-bg-tag-accent, #u-color-text-dark

Mathematics is full of wonderful things, but nothing strikes me as more wonderful—or more unintuitive—than Gabriel's Horn.

Here's how it works: take the function \\(y = \\frac{1}{x}\\) where \\(x \\in \\mathbb{R}^+, 1 \\leq x \\leq \\infty\\), and rotate it around the \\(x\\) axis. Not too hard to picture—it looks like a horn. But here's where it gets weird.

Let's calculate the volume. Using solids of revolution, we can show:

$$  
V = \\pi \\lim\_{t \\rightarrow \\infty} \\int \_1 ^t \\frac{1}{x^2} dx = \\pi \\lim \_{t \\rightarrow \\infty} ( 1 - \\frac{1}{t} ) = \\pi  
$$

Simple, elegant. The volume is exactly \\(\\pi\\). Now let's check the surface area.

We know the general definition of arc length is \\(\\int \_a ^b \\sqrt{1 + f'(x)^2}\\). Combining this with our solids of revolution gives us:

$$  
A = 2\\pi \\lim \_{t \\rightarrow \\infty} \\int \_1 ^t \\frac{1}{x} \\sqrt{1 + \\left( -\\frac{1}{x^2} \\right)^2 } dx  
$$

This integral isn't trivial, but there's a trick. Take the simpler integral $$2\\pi \\lim \_{t \\rightarrow \\infty} \\int \_1 ^t \\frac{dx}{x}$$ instead. This will always be less than or equal to our original integral (we dropped the \\(\\sqrt{1 + (-\\frac{1}{x^2})^2}\\) term, which is always ≥ 1). Evaluating this simpler integral:

$$  
A \\geq 2\\pi \\lim \_{t \\rightarrow \\infty} \\int \_1 ^t \\frac{dx}{x} \\implies A \\geq \\lim \_{t \\rightarrow \\infty} 2\\pi \\ln(t)  
$$

Wait. It diverges. The volume is \\(V = \\pi\\), but the surface area is \\(A \\geq \\infty\\). This isn't a mistake—the math checks out. And that's simply wonderful.

![](https://starikov.co/content/images/2025/11/gabriels_horn_visualization.png)

*A horn you can fill with paint, but you can't paint the surface.*